Chapter4.ver5c.p.87

Chapter4.ver5c.p.87 - Section 4.9 Laplaces equation in...

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Section 4.9 Laplace’s equation in cylindrical coordinates As in the case of spherical coordinates, this equation is solved by a series expansion in terms of products of functions of the individual cylindrical coordinates. That is, we use separation of variables. Φ ( ρ, ϕ, z )= X l,m,n α lmn U l ( ρ ) V m ( ϕ ) W k ( z ) . (4.53) Substituting Φ lmk ( ρ, ϕ, z U l ( ρ ) V m ( ϕ ) W k ( z ) . into Eq 4.52 and dividing by Φ lmk ( ρ, ϕ, z ) , we can sequentially separate all the variable .dependence: 1 U l ρ d · ρ d U l ¸ + 1 V m ρ 2 d 2 2 V m = 1 W n d 2 dz 2 W k = ± k 2 (4.54) ρ U l d · ρ d U l ¸ ρ 2 k 2 = 1 V m d 2 2 V m = ± m 2 (4.55) These functions satisfy the Eqs. 4.54, 4.55 and 4.56: d 2 W k ( z ) dz 2 = λ z W k ( z ) ; λ z = k 2 (4.54a) W k ( z A k e kz + B k e kz (4.54b) Note that k can be complex, pure imaginary or real. d 2 2 V m ( ϕ λ ϕ V m ( ϕ ) ; λ ϕ = m 2 (4.55) V m ( ϕ C m e imϕ + D m e imϕ (4.55b) Note that m could be non-integer and/or complex ρ 2 d 2 2 U ( ρ )+ ρ d U ( ρ ¡ k 2 ρ 2 m 2 ¢ U ( ρ )=0 (4.56) ( ) 2 d 2 d ( ) 2 U m ( d d ( ) U m ( ¡ ( ) 2 m 2 ¢ U m ( U m ( a m J m ( b m N m ( ) (4.56b) wherenowthelabelon U m ( ) becomes m and the J m and . N m are called Bessel functions of the
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